LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034
M.Sc. DEGREE EXAMINATION – MATHEMATICS
FIRST SEMESTER – APRIL 2008
XZ 28
MT 1807 - DIFFERENTIAL GEOMETRY
Date : 05-05-08
Time : 1:00 - 4:00
Dept. No.
Max. : 100 Marks
Answer all the questions
I a) Obtain the equation of the tangent at any point on the circular helix.
(or)
b) Prove that the curvature is the rate of change of angle of contingency with respect to
arc length.
[5]
c) Derive the formula for torsion of a curve in terms of the parameter u and hence
calculate the torsion and curvature of the curve x  u, y  u 2 and z  u 3 .
(or)
d) Derive the Serret-Frenet formulae and deduce them in terms of Darboux vector.[15]
II a) If the curve lx  my  nz  0, x 2  y 2  z 2  2cz has three point contact with origin
l 2  m2
.
bl 2  am2
(or)
b) Prove that the necessary and sufficient condition that a space curve may be helix is
that the ratio of its curvature to torsion is always a constant.
[5]
with cx 2  by 2  cz then prove that c 
c) Define evolute and involute. Also find their equations.
(or)
d) State and prove the fundamental theorem of space curves.
[15]
III a) Derive the equation satisfying the principal curvature at a point on the space curve.
(or)
b) Prove that the first fundamental form is positive definite.
[5]
c) Prove the necessary and sufficient condition for a surface to be developable.
(or)
d) Derive any two developables associated with a space curve.
IV a) State the duality between space curve and developable.
(or)
b) Derive the geometrical interpretation of second fundamental form.
c) Find the first and second fundamental form of the curve
x  a cos  sin  , y  a sin  sin  , z  a cos  .
(or)
d) Find the principal curvature and direction of the surface
x  u cos v, y  u sin v, z  av .
[15]
[5]
[15]
V a) Derive Weingarton equation.
(or)
b) Show that sphere is the only surface in which all points are umbilics.
[5]
c) Derive Gauss equation in terms of Christoffel’s symbol.
(or)
(d) State the fundamental theorem of Surface Theory and demonstrate it in the case
of unit sphere .
[15]
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